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Tuesday, January 26, 2010

Mike South on The Meaning of Zero

I'm on a google group called NaturalMath, hosted by Maria Droujkova. We were discussing how it can take some deeper thinking to understand zero.

One person wrote:

Maybe because we use [zero] as a symbol for the abstract concept of nothing, which none of us have experience with, it prevents other considerations...

Mike South's reply struck me, and I asked him if I could report it here. So you guest blogger for today is Mike South...

Discovering Zero

I don't know--we don't have experience with circles, either--not perfect ones--but I think we can imagine them pretty well.

I can ask myself "how man people are in the room besides me?" and want to have something to call that when the last one leaves. But I think that particular idea is post hoc. I think people discover zero when they realize they need it for something. Similar to negative numbers--not something people (generally) sit around and think up, but they come up naturally.

I would guess that most of the times people discover a need for zero (not that we let them do that any more, preferring to push it on them when we think it's time) it comes in a positional number system like the dude on the wikipedia page who just kind of "needed something to keep the other numbers lined up". Eventually they figure out that there is a perfectly logical way of thinking about it as a "real" entity and not just a made-up mental crutch to get you to the "real" answer.

Time is a possible example where you could start thinking of zero as something real. You can represent two days from now as 2, 1 day from now as 1, 1 day ago as -1, 2 days ago as -2, and then say, well, what about today? When it's "no days ago" or "no days from now", what do we write for that? You can say "well, that's just today", and I think people probably normally start there. But eventually it creeps into your mind that maybe that ought to be a number in its own right.

European-style floor numbering is another good example, I think. In (at least some) countries in Europe, they number the floors based on how far up you are from street level. So what we would call the second floor, they would call something like "one floor up". And if you number one floor up on the elevator buttons as "1" (and they do), and 2 floors up as 2, well, one floor down (say you have a basement parking garage) is -1, 2 floors down is -2, what do you label the button for the ground floor? You could put a "G" or "M" for main floor, but that should eventually get people thinking--why no number for the threshold between up and down? You can probably go for a long time saying "it's not up or down. It's just ground level, and that's why we call it that--you don't need a number because there is no distance." And in some sense that's simply correct.

But once you let that little fellow creep in to your mathematical vocabulary, it ends up being indispensable and making things way, way easier to deal with. And that is probably ultimately what ends up being thought of as "real"--if it's useful in many situations, if we are mentally mapping either abstract problems or physical realities onto it, we end up wanting it into existence.

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Mike South has written a few more things you might like, available at his Fulcrum site.

About Me

Math Mama is Sue VanHattum, a community college math teacher interested in all levels of math learning, and the mama of a young son. I entered the blogging world as I began work on an anthology about learning math. Contact me at mathanthologyeditor on gmail etc.